Calculus 3 Exam 1

The general equation x²/a²+ y²/b²+ z²/c²=1 represents?

An ellipsoid
The general equation z/c = x²/a²+y²/b²represents?

Elliptic paraboloid
The general equation z²/c²= x²/a²+ y²/b²represents?

A cone
The general equation z/c = x²/a² - y²/b² represents?

A hyperbolic paraboloid
The general equation - x²/a²- y²/b²+ z²/c²= 1 represents?

A hyperboloid of 2 sheets
The general equation x²/a²+ y²/b²- z²/c²= 1 represents?

Hyperboloid of 1 sheet
How do you add and subtract vectors?

Place tip-to-tail in their respected directions and draw a new vector originating from the first vector's origin.
How do you find th vector between two points?

Take the new components minus the old components.
When are two vectors orthogonal?

If and only if their dot product = 0.
How do you find the angle between two vectors?

Cos() = a b / |a| |b|
What are the direction cosines?
- Cos() = a₁/ |a|
- Cos() = a₂/ |a|
- Cos() = a₃ / |a| a is the vector itself
What is the scalar projection?

Comp_ab = a b / |a|
What is the vector projection?

Proj_ab =( a b / |a|² )a
How do you find the magnitude of a vector?

(X)² + (y)²+ (z)²
a X b is _________ to both a and b.

Orthogonal
When are two vectors parallel?

Ifand only if their cross product = 0.
How do you find the area of a parallelogram created by two vertices?

Find the cross product of them, then take the magnitude of it.